Functional determinants by contour integration methods

TL;DR

This paper introduces a straightforward contour integration approach to derive formulas for functional determinants, including cases with zero modes, enhancing analytical tools in mathematical physics.

Contribution

It presents a new, accessible method for calculating functional determinants applicable to various differential operators, including those with zero modes.

Findings

Derived explicit formulas for functional determinants using contour integration.

Extended the method to cases with zero modes, which are physically significant.

Demonstrated the method on second order differential operators with Dirichlet boundary conditions.

Abstract

We present a simple and accessible method which uses contour integration methods to derive formulae for functional determinants. To make the presentation as clear as possible, the general idea is first illustrated on the simplest case: a second order differential operator with Dirichlet boundary conditions. The method is applicable to more general situations, and we discuss the way in which the formalism has to be developed to cover these cases. In particular, we also show that simple and elegant formulae exist for the physically important case of determinants where zero modes exist, but have been excluded.